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Calculus

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CliffsQuickReviewTM

Calculus

By Bernard V. Zandy, MA and

Jonathan J. White, MS

An International Data Group Company

New York, NY • Cleveland, OH • Indianapolis, IN

6376-9 FM.F 4/25/01 8:48 AM Page i

CliffsQuickReview™Calculus

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About the Authors

Bernard V. Zandy, MA, Professor of Mathematics

at Fullerton College in California has been teach-

ing secondary and college level mathematics for

34 years. A co-author of the Cliffs PSAT and SAT

Preparation Guides, Mr. Zandy has been a lecturer

and consultant for Bobrow Test Preparation Ser-

vices, conducting workshops at California State

University and Colleges since 1977.

Jonathan J. White has a BA in mathematics from

Coe College and an MS in mathematics from the

University of Iowa. He is currently pursuing a

PhD in Mathematics Pedagogy and Curriculum

Research at the University of Oklahoma.

Publisher’s Acknowledgments

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6376-9 FM.F 4/25/01 8:48 AM Page ii

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Visit Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Chapter 1: Review Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Interval Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

Chapter 2: Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Intuitive Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Evaluating Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

One-sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Limits Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .23

Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

Chapter 3: The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Differentation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

Trigonometric Function Differentiation . . . . . . . . . . . . . . . . . . . . . . .34

Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

Differentiation of Inverse Trigonometric Functions . . . . . . . . . . . . . .40

Differentiation of Exponential and Logarithmic Functions . . . . . . . .41

Chapter 4: Applications of the Derivative . . . . . . . . . . . . . . . . . . . . . . . .43

Tangent and Normal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

Increasing/Decreasing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .48

First Derivative Test for Local Extrema . . . . . . . . . . . . . . . . . . . . . . . .49

Second Derivative Test for Local Extrema . . . . . . . . . . . . . . . . . . . . .50

Concavity and Points of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . .51

Maximum/Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

Distance, Velocity, and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .55

Related Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

6376-9 FM.F 4/25/01 8:48 AM Page iii

Chapter 5 : Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

Antiderivatives/Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . .63

Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

Basic formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

Substitution and change of variables . . . . . . . . . . . . . . . . . . . . . . .66

Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

Trigonometric integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

Distance, Velocity, and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .73

Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

Definition of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .75

Properties of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . .80

Definite integral evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

Chapter 6: Applications of the Definite Integral . . . . . . . . . . . . . . . . . .88

Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88

Volumes of Solids with Known Cross Sections . . . . . . . . . . . . . . . . . .93

Volumes of Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

Disk method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

Washer method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97

Cylindrical shell method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99

Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113

Appendix: Using Graphing Calculators in Calculus . . . . . . . . . . . . . . .116

Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117

Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121

iv CliffsQuickReview Calculus

6376-9 FM.F 4/25/01 8:48 AM Page iv

Introduction

Calculus is the mathematics of change. Any situation that involves quan-

tities that change over time can be understood with the tools of cal-

culus. Differential calculus deals with rates of change or slopes, and is

explored in Chapters 3 and 4 of this book. Integral calculus handles total

changes or areas, and is addressed in Chapters 5 and 6. Although it is not

always immediately obvious, this mathematical notion of change is essen-

tial to many areas of knowledge, particularly disciplines like physics, chem-

istry, biology, and economics.

The prerequisites for learning calculus include much of high school alge-

bra and trigonometry, as well as some essentials of geometry. If the for-

mulas on the front side of the Pocket Guide (the cardstock page right inside

the front cover) and topics covered in Chapter 1 are familiar to you, then

you probably have sufficient background to begin learning calculus. If

some of those are unfamiliar, or just rusty for you, then CliffsQuickReview

Geometry, CliffsQuickReview Algebra, or CliffsQuickReview Trigonometry

may be valuable starting points for you.

Why You Need This Book

Can you answer yes to any of these questions?

■Do you need to review the fundamentals of calculus fast?

■Do you need a course supplement to calculus?

■Do you need a concise, comprehensive reference for calculus?

If so, then CliffsQuickReview Calculus is for you!

How to Use This Book

You can use this book in any way that fits your personal style for study and

review—you decide what works best with your needs. You can either read

the book from cover to cover or just look for the information you want

and put it back on the shelf for later. Here are just a few ways you can use

this book:

■Read the book as a stand-alone textbook to learn all the major con-

cepts of calculus.

6376-9 Intro.F 4/25/01 8:48 AM Page 1

■Use the Pocket Guide to find often-used formulas, from calculus and

other relevant formulas from algebra, geometry and trigonometry.

■Refer to a single topic in this book for a concise and understandable

explanation of an important idea.

■Get a glimpse of what you’ll gain from a chapter by reading through

the “Chapter Check-In” at the beginning of each chapter.

■Use the Chapter Checkout at the end of each chapter to gauge your

grasp of the important information you need to know.

■Test your knowledge more completely in the CQR Review and look

for additional sources of information in the CQR Resource Center.

■Review the most important concepts of an area of calculus for an

exam.

■Brush up on key points as preparation for more advanced mathe-

matics.

Being a valuable reference source also means it’s easy to find the informa-

tion you need. Here are a few ways you can search for topics in this book:

■Look for areas of interest in the book’s Table of Contents, or use the

index to find specific topics.

■Use the glossary to find key terms fast. This book defines new terms

and concepts where they first appear in the chapter. If a word is bold-

faced, you can find a more complete definition in the book’s glossary.

■Flip through the book looking for subject areas at the top of each

page.

■Or browse through the book until you find what you’re looking

for—we organized this book to gradually build on key concepts.

Visit Our Web Site

A great resource, www.cliffsnotes.com features review materials,

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2CliffsQuickReview Calculus

6376-9 Intro.F 4/25/01 8:48 AM Page 2

Chapter 1

REVIEW TOPICS

Chapter Check-In

❑Reviewing functions

❑Using equations of lines

❑Reviewing trigonometric functions

Certain topics in algebra, geometry, analytical geometry, and trigonom-

etry are essential in preparing to study calculus. Some of them are

briefly reviewed in the following sections.

Interval Notation

The set of real numbers (R) is the one that you will be most generally con-

cerned with as you study calculus. This set is defined as the union of the

set of rational numbers with the set of irrational numbers. Interval nota-

tion provides a convenient abbreviated notation for expressing intervals of

real numbers without using inequality symbols or set-builder notation.

The following lists some common intervals of real numbers and their

equivalent expressions, using set-builder notation:

, : < <a b x R a x b!=

^ h

" ,

, :a b x R a x b! # #=

" ,

[ , ,)

: <a b x R a x b! #=

" ,

( , ] : <a b x R a x b! #=

" ,

, : >a x R x a3!+ =

^ h

" ,

[ , ) :a x R x a3! $+ =

" ,

, : <b x R x b3!- =

^ h

" ,

6376-9 Ch01.F 4/25/01 8:48 AM Page 3

( , ] :b x R x b3! #- =

" ,

,x R3 3 !- + =

^ h

" ,

Note that an infinite end point !3

^ h

is never expressed with a bracket in

interval notation because neither 3+nor 3-represents a real number

value.

Absolute Value

The concept of absolute value has many applications in the study of cal-

culus. The absolute value of a number x, written xmay be defined in a

variety of ways. On a real number line, the absolute value of a number is

the distance, disregarding direction, that the number is from zero. This

definition establishes the fact that the absolute value of a number must

always be nonnegative—that is, x0$.

A common algebraic definition of absolute value is often stated in three

parts, as follows:

, >

, <

x x

0 0

= =

[

]

Another definition that is sometimes applied to calculus problems is

x x 2

or the principal square root of x2. Each of these definitions also implies

that the absolute value of a number must be a nonnegative.

Functions

A function is defined as a set of ordered pairs (x,y), such that for each first

element x, there corresponds one and only one second element y. The set

of first elements is called the domain of the function, while the set of sec-

ond elements is called the range of the function. The domain variable is

referred to as the independent variable, and the range variable is referred

to as the dependent variable. The notation f(x) is often used in place of y

to indicate the value of the function ffor a specific replacement for xand

is read “f of x” or “f at x.”

4CliffsQuickReview Calculus

6376-9 Ch01.F 4/25/01 8:48 AM Page 4

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